Summary form only given. We investigate the role of the collective symmetric and antisymmetric states in entanglement creation by spontaneous emission in a system of two non-overlapping two-level atoms. We calculate and illustrate graphically populations of the collective atomic states and the Wootters entanglement measure (concurrence) for the atoms prepared in different initial states: for only one atom excited and both atoms excited. Our calculations include the dipole-dipole interaction and a spatial separation between the atoms that the antisymmetric state of the system is included throughout even for small interatomic separations. It is shown that spontaneous emission can lead to a transient entanglement between the atoms even if the atoms were prepared initially in an unentangled state. We find that the ability of spontaneous emission to create the transient entanglement relies on the absence of population in the collective symmetric state of the system. For an initial state of only one atom prepared in the excited state, the entanglement builds up rapidly in time and reaches a maximum for the parameter values corresponding roughly to zero population in the symmetric state. On the other hand, for the initial condition of both atoms in their excited states, the system does not evolve into an entangled state until the symmetric state, is depopulated. We also study entanglement creation in a system of two non-identical atoms of different transition frequencies and find that the entanglement between the atoms can be enhanced compared to that for identical atoms. In addition, we show that the entanglement can decay with two different time scales resulting from the coherent transfer of the population from the symmetric to the antisymmetric state.